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18.
2.2.
Tensorial
Representaon
• Adjacency
tensor
for
unweighted
case:
• Elements
of
adjacency
tensor:
– Auvαβ
=
Auvα1β1
…
αdβd
=
1
iff
((u,α),
(v,β))
is
an
element
of
EM
(else
Auvαβ
=
0)
• Important
note:
‘padding’
layers
with
empty
nodes
– One
needs
to
disnguish
between
a
node
not
present
in
a
layer
and
nodes
exisng
but
edges
not
present
(use
a
supplementary
tensor
with
labels
for
edges
that
could
exist),
as
this
is
important
for
normalizaon
in
many
quanes.

24.
%NCUUKH[KPI/WNVKNC[GT0GVYQTMU
• Special
cases
of
mullayer
networks
include:
mulplex
networks,
interdependent
networks,
networks
of
networks,
node-­‐colored
networks,
edge-­‐colored
mulgraphs,
…
• To
obtain
one
of
these
special
cases,
we
impose
constraints
on
the
general
structure
defined
earlier.
• See
the
review
arcle
for
details.

25.
%QPUVTCKPVU
HTQOVJG6CDNG
• 1.
Node-­‐aligned
(or
fully
interconnected):
All
layers
contain
all
nodes.
• 2.
Layer
disjoint:
Each
node
exists
in
at
most
one
layer.
• 3.
Equal
size:
Each
layer
has
the
same
number
of
nodes
(but
they
need
not
be
the
same
ones).
• 4.
Diagonal
coupling:
Inter-­‐layer
edges
only
can
exist
between
nodes
and
their
counterparts.
• 5.
Layer
coupling:
coupling
between
layers
is
independent
of
node
identy
– Note:
special
case
of
“diagonal
coupling”
• 6.
Categorical
coupling:
diagonal
couplings
in
which
inter-­‐layer
edges
can
be
present
between
any
pair
of
layers
– Contrast:
“ordinal”
coupling
for
tensorial
representaon
of
temporal
networks
• Example
1:
Most
––
but
not
all!
––
“mul@plex
networks”
studied
in
the
literature
sasfy
(1,3,4,5,6)
and
include
d
=
1
aspects.
– Note:
Many
important
situaons
need
(1,3)
to
be
relaxed.
(E.g.
Some
people
have
Facebook
accounts
but
not
Twi;er
accounts.)
• Example
2:
The
“networks
of
networks”
that
have
been
invesgated
thus
far
sasfy
(3)
and
have
addional
constraints
(which
can
be
relaxed).

31.
/WNVKRNGZ0GVYQTMU
• Networks
with
multiple
types
of
edges
– Also
known
as
multirelational
networks,
edge-­‐colored
multigraphs,
etc.
• Many
studies
in
practice
use
the
same
sets
of
nodes
in
each
layer,
but
this
isn’t
required.
– Challenge
for
tensorial
representation:
need
to
keep
track
of
lack
of
presence
of
a
tie
versus
a
node
not
being
present
in
a
layer
(relevant
e.g.
for
normalization
of
multiplex
clustering
coefficients)
• Question:
When
should
you
include
inter-­‐
layer
edges
and
when
should
you
ignore
them?

32.
*[RGTITCRJU
• Hyperedges
generalize
edges.
A
hyperedge
can
include
any
(nonzero)
number
of
nodes.
• Example:
A
k-­‐uniform
hypergraph
has
cardinality
k
for
each
hyperedge
(e.g.
a
folksonomy
like
Flickr).
– One
can
represent
a
k-­‐uniform
hypergraph
using
adjacency
tensors,
and
there
have
been
some
studies
of
multiplex
networks
by
mapping
them
into
k-­‐uniform
hypergraphs.
– A
nice
paper:
Michoel
Nachtergaele,
PRE,
2012
• Note
that
multilayer
networks
are
still
formulated
for
pairwise
connections
(but
a
more
general
type
of
pairwise
connections
than
usual).

33.
1TFKPCN%QWRNKPIUCPF
6GORQTCN0GVYQTMU
• Ordinal
coupling:
diagonal
inter-­‐layer
edges
among
consecutive
layers
(e.g.
multilayer
representation
of
a
temporal
network)
• Categorical
coupling:
diagonal
inter-­‐layer
edges
between
all
pairs
of
edges
• Both
can
be
present
in
a
multilayer
network,
and
both
can
be
generalized

34.
1VJGT6[RGUQH
/WNVKNC[GT0GVYQTMU
GZCORNGU
• k-­‐partite
graphs
– Bipartite
networks
are
most
commonly
studied
• Coupled-­‐cell
networks
– Associate
a
dynamical
system
with
each
node
of
a
multigraph.
Network
structure
through
coupling
terms.
• Multilevel
networks
– Very
popular
in
social
statistics
literature
(upcoming
special
issue
of
Social
Networks)
– Each
level
is
a
layer
– Think
‘hierarchical’
situations.
Example:
‘micro-­‐
level’
social
network
of
researchers
and
a
‘macro-­‐
level’
for
a
research-­‐exchange
network
between
laboratories
to
which
the
researchers
belong

39.
2TCEVKECNKVKGUCPF/GUUCIGU
• Lots
of
reliable
data
on
intra-­‐layer
relations
(i.e.
the
usual
kind
of
edges)
• It’s
much
more
challenging
to
collect
reliable
data
for
inter-­‐layer
edges.
We
need
more
data.
– E.g.
Transportation
data
should
be
a
very
good
resource.
Think
about
the
amount
of
time
to
change
gates
during
a
layover
in
an
airport.
– E.g.
Transition
probabilities
of
a
person
using
different
social
media
(each
medium
is
a
layer).
• Most
empirical
multilayer-­‐network
studies
thus
far
have
tended
to
be
multiplex
networks.
• Determining
inter-­‐layer
edges
as
a
problem
in
trying
to
reconcile
node
identities
across
networks.
(Can
you
figure
out
that
a
Twitter
account
and
Facebook
account
belong
to
the
same
person?)
– Major
implications
for
privacy
issues
• Take-­‐home
message:
Be
creative
about
how
you
construct
multilayer
networks
and
define
layers!

42.
#IITGICVKQPQH
/WNVKNC[GT0GVYQTMU
• Construct
single-­‐layer
(i.e.
“monoplex”)
networks
and
apply
the
usual
tools.
– Obtain
edge
weights
as
weighted
average
of
connections
in
different
layers.
You
get
a
different
weighted
network
with
a
different
weighting
vector.
• E.g.
Zachary
Karate
Club
– Information
loss
• Is
there
a
way
to
do
this
to
minimize
information
loss?
• Important:
Loss
of
“Markovianity”
(a
la
temporal
networks)
– Processes
that
are
Markovian
on
a
multilayer
network
may
yield
non-­‐Markovian
processes
after
aggregating
the
network

43.
KCIPQUVKEU
• Generalizations
of
the
usual
suspects
– Degree/strength
– Neighborhood
• Which
layers
should
you
consider?
– Centralities
– Walks,
paths,
and
distances
– Transitivity
and
local
clustering
• Important
note:
Sometimes
you
want
to
define
different
values
for
different
node-­‐layers
(e.g.
a
vector
of
centralities
for
each
entity)
and
sometimes
you
want
a
scalar.
• Need
to
be
able
to
consider
different
subsets
of
the
layers
• Need
more
genuinely
multilayer
diagnostics
– It
is
important
to
go
beyond
“bigger
and
better”
versions
of
the
usual
concepts.

44.
GITGGUCPF0GKIJDQTJQQFU
• Simplest
way:
Use
aggregation
and
then
measure
degree,
strength,
and
neighborhoods
on
a
monoplex
network
obtained
from
aggregation.
– Possibly
only
consider
a
subset
of
the
layers
• More
sophisticated:
Define
a
multi-­‐edge
as
a
vector
to
track
the
information
in
each
layer.
With
weighted
multilayer
networks,
you
can
keep
track
of
different
weights
in
intra-­‐layer
versus
inter-­‐layer
edges.
• Towards
multilayer
measures:
overlap
multiplicity
for
a
multiplex
network
can
track
how
often
an
edge
between
entities
i
and
j
occurs
in
multiple
layers

48.
9CNMU2CVJUCPFKUVCPEGU
• To
define
a
walk
(or
a
path)
on
a
multilayer
network,
we
need
to
consider
the
following:
– Is
changing
layers
considered
to
be
a
step?
Is
there
a
“cost”
to
changing
layers?
How
do
you
measure
this
cost?
• E.g.
transportation
networks
vs
social
networks
– Are
intra-­‐layer
steps
different
in
different
layers?
• Example:
labeled
walks
(i.e.
compound
relations)
are
walks
in
a
multiplex
network
that
are
associated
with
a
sequence
of
layer
labels
• Generalizing
walks
and
paths
is
necessary
to
develop
generalizations
for
ideas
like
clustering
coefficients,
transitivity,
communicability,
random
walks,
graph
distance,
connected
components,
betweenness
centralities,
motifs,
etc.
• Towards
multilayer
measures:
Interdependence
is
the
ratio
of
the
number
of
shortest
paths
that
traverse
more
than
one
layer
to
the
number
of
shortest
paths

49.
%NWUVGTKPI%QGHHKEKGPVU
CPF6TCPUKVKXKV[
• Our
approach:
Cozzo
et
al.,
2013
– Use
the
idea
of
multilayer
walks.
Keep
track
of
returning
to
entity
i
(possibly
in
a
different
layer
from
where
we
started)
separately
for
1
total
layer,
2
total
layers,
3
total
layers
(and
in
principle
more).
• Insight:
Need
different
types
of
transitivity
for
different
types
of
multiplex
networks.
– Example
(again):
transportation
vs
social
networks
– There
are
several
different
clustering
coefficients
for
monoplex
weighted
networks,
and
this
situation
is
even
more
extreme
for
multilayer
networks.

50.
'ZCORNG%NWUVGTKPI%QGHHKEKGPV
%QQGVCNCT:KX
• Our perspective:
multilayer walks,
which can return
to node i on
different layers
and traverse
different numbers
of layers!

51.
%GPVTCNKV[/GCUWTGU
• In
studies
of
networks,
people
compute
a
crapload
of
centralities.
• The
common
ones
have
been
generalized
in
various
ways
for
multilayer
networks.
– Again,
one
needs
to
ask
whether
you
want
a
centrality
for
a
node-­‐layer
or
for
a
given
entity
(across
all
layers
or
a
subset
of
layers).
• Eigenvector
centralities
and
related
ideas
can
be
derived
from
random
walks
on
multilayer
networks.
– Consider
different
spreading
weights
for
different
types
of
edges
(e.g.
intra-­‐layer
vs
inter-­‐layer
edges;
or
different
in
different
layers)
• Betweenness
centralities
can
be
calculated
for
different
generalizations
of
short
paths.
• A
point
of
caution:
“What
the
world
needs
now
is
another
centrality
measure.”
– I.e.
although
they
can
be
very
useful,
please
don’t
go
too
crazy
with
them.

52.
+PVGTNC[GTKCIPQUVKEU
• The
community
needs
to
construct
genuinely
multilayer
diagnostics
and
go
beyond
‘bigger
and
better’
versions
of
the
concepts
we
know
and
(presumably)
love.
– Not
very
many
yet
• Correlations
of
network
structures
between
layers
– E.g.
interlayer
degree-­‐degree
correlations
(or
any
other
diagnostic)
• !
Interpreting
communities
as
layers,
quantities
like
assortativity
can
be
construed
as
inter-­‐layer
diagnostics
• Interdependence
is
the
ratio
of
the
number
of
shortest
paths
that
traverse
more
than
one
layer
to
the
number
of
shortest
paths

54.
/QFGNUQH/WNVKRNGZ0GVYQTMU
• Statistical-­‐mechanical
ensembles
of
multiplex
networks
• Generalize
growth
mechanisms
like
preferential
attachment
– Again,
one
can
include
inter-­‐layer
correlations
in
designing
a
model
• It
would
be
good
to
go
beyond
“bigger
and
better”
versions
of
the
usual
ideas.
– Including
simple
inter-­‐layer
correlations
(especially
between
intra-­‐
layer
degrees)
has
been
the
main
approach
so
far.

55.
/QFGNUQH
+PVGTEQPPGEVGF0GVYQTMU
• Straightforward:
Construct
different
layers
separately
using
your
favorite
model
(or
even
one
that
you
hate)
and
then
add
inter-­‐layer
edges
uniformly
at
random.
• More
sophisticated:
Be
more
strategic
in
adding
inter-­‐layer
edges.
• Some
random-­‐graph
modules
with
community
structure
can
be
useful,
where
we
think
of
each
community
as
a
separate
layer
(i.e.
as
a
separate
network
in
a
network
of
networks)
– E.g.
Melnik
et
al’s
paper
(Chaos,
2014)
on
random
graphs
with
heterogeneous
degree
assortativity
• The
homophily
is
different
in
different
layers
and
there
is
a
mixing
matrix
for
inter-­‐layer
connections

56.
%QOOWPKVKGUCPF1VJGT
/GUQUECNG5VTWEVWTGU
• Communities
are
dense
sets
of
nodes
in
a
network
(typically
relative
to
some
null
model).
– One
can
use
these
ideas
for
multilayer
networks
(e.g.
multislice
modularity).
• Interpreting
communities
as
roadblocks
to
some
dynamical
process
(e.g.
starting
from
some
initial
condition),
one
can
have
such
a
process
on
a
multilayer
network—with
different
spreading
rates
in
different
types
of
edges—to
algorithmically
find
communities
in
multilayer
networks.
• Most
work
thus
far
on
multilayer
representation
of
temporal
networks.
– One
exception
is
recent
work
on
“Kantian
fractionalization”
in
international
relations.
• Challenge:
Develop
multilayer
null
models
for
community
detection
(different
for
ordinal
vs.
categorical
coupling)
• Blockmodels
• Spectral
clustering
(e.g.
Michoel
Nachtergaele)
• Note:
Because
I
have
done
a
lot
of
work
in
this
area,
I
will
go
through
a
bit
in
some
detail
to
help
illustrate
some
general
points
that
are
also
relevant
in
other
studies
of
multilayer
networks.

57.
! Communities = Cohesive
groups/modules/
mesoscopic structures
› In stat phys, you try to
derive macroscopic and
mesoscopic insights from
microscopic information
! Community structure
consists of complicated
interactions between
modular (horizontal)
and hierarchical
(vertical) structures
! communities have denser
set of Internal edges
relative to some null
model for what edges
are present at random
› “Modularity”

65.
[PCOKE4GEQPHKIWTCVKQPQH*WOCP
$TCKP0GVYQTMUWTKPI.GCTPKPIa
$CUUGVVGVCN20#5
• fMRI
data:
network
from
correlated
time
series
• Examine
role
of
modularity
in
human
learning
by
identifying
dynamic
changes
in
modular
organization
over
multiple
time
scales
• Main
result:
flexibility,
as
measured
by
allegiance
of
nodes
to
communities,
in
one
session
predicts
amount
of
learning
in
subsequent
session

66.
Staonarity
and
Flexibility
• Community
staonarity
ζ
(autocorrelaon
over
me
of
community
membership):
• Node
flexibility:
– fi
=
number
of
mes
node
i
changed
communies
divided
by
total
number
of
possible
changes
– Flexibility
f
=
fi

67.
[PCOKE%QOOWPKV[5VTWEVWTG
• Investigating
community
structure
in
a
multilayer
framework
requires
consideration
of
new
null
models
• Many
more
details!
– E.g.,
Robustness
of
results
to
choice
of
size
of
time
window,
size
of
inter-­‐slice
coupling,
particular
definition
of
flexibility,
complicated
modularity
landscape,
definition
of
‘similarity’
of
time
series,
etc.

68.
Dynamic
Reconfiguraon
of
Human
Brain
Networks
During
Learning
(Basse;
et
al,
PNAS,
2011)
• fMRI
data:
network
from
correlated
me
series
• Examine
role
of
modularity
in
human
learning
by
idenfying
dynamic
changes
in
modular
organizaon
over
mulple
me
scales
• Main
result:
flexibility,
as
measured
by
allegiance
of
nodes
to
communies,
in
one
session
predicts
amount
of
learning
in
subsequent
session

70.
/GVJQFU$CUGFQP
6GPUQTGEQORQUKVKQP
• Many
different
generalizations
of
singular
value
decomposition
(SVD)
to
tensors
– Every
matrix
has
a
unique
SVD,
but
we
have
to
relax
this
for
tensors.
– See
Kolda
and
Bader,
SIAM
Review,
2009
– Tensor
rank
vs
matrix
rank:
hard
to
determine
that
rank
of
tensors
of
order
3+
• Note:
“rank”
is
also
used
as
a
synonym
for
“order”
(see
earlier).
Here,
“rank”
is
the
generalization
of
matrix
rank:
the
minimum
number
of
column
vectors
needed
to
span
the
range
of
a
matrix.
The
tensor
rank
is
the
minimum
number
of
rank-­‐1
tensors
with
which
one
can
express
a
tensor
as
a
sum.
The
purpose
of
an
SVD
(and
generalizations)
is
to
find
a
low-­‐rank
approximation.
• Non-­‐negative
tensor
factorization

72.
%QPPGEVGF%QORQPGPVU
CPF2GTEQNCVKQP
• Connected
component
defined
as
in
monoplex
networks,
except
that
multiple
types
of
edges
can
occur
in
a
path.
• In
multilayer
networks,
one
again
uses
branching-­‐process
approximations
that
allow
the
use
of
generating
function
technology.
– Same
fundamental
idea
(and
limitations)
as
in
monoplex
networks,
but
the
calculations
are
more
intricate
• More
flavors
of
giant
connected
components
(GCCs)
that
can
be
defined

80.
%QWRNGFEGNN0GVYQTMU
• Each
node
is
associated
with
a
dynamical
system,
and
two
nodes
have
the
same
color
if
they
have
the
same
state
space
and
an
identical
dynamical
system.
• The
couplings
between
dynamical
systems
are
the
edges
(or
hyperedges).
Two
edges
have
the
same
color
if
the
couplings
are
equivalent
• There
exist
many
nice
results
for
generic
bifurcations
in
small
coupled-­‐celled
networks.
– Spiritually
similar
results
for
generic
phase
transitions
in
random
walks
and
Laplacians,
but
for
very
low-­‐dimensional
systems
instead
of
high-­‐
dimensional
ones
• Surgeon
General’s
warning:
The
papers
on
coupled-­‐
cell
networks
(many
by
Marty
Golubitsky
and
company)
are
very
mathematical.

82.
%QPVTQNCPF[PCOKEU
• It’s
important
to
consider
feedback
loops.
• Maybe
one
is
only
allowed
to
apply
controls
to
a
subset
of
the
layers?
• Layer
decompositions:
Start
with
a
network
and
try
to
infer
layers
– Reminiscent
of
community
detection,
but
with
layers
instead
of
dense
modules
– E.g.
research
by
Prescott
and
Papachristodoulou
on
biochemical
networks
– Similar
problem
in
social
networks
• “Control
network”
used
to
influence
an
“open-­‐loop
network”
(which
doesn’t
include
feedback)
• “Pinning
control”,
in
which
one
controls
a
small
fraction
of
nodes
to
try
to
influence
the
dynamics
of
other
nodes,
in
the
context
of
interconnected
networks.

85.
“What
befell
Candide
us
at
the
end
of
his
our
journey”
%10%.75+105#0
176.11-

86.
%QPENWUKQPU
• Multilayer networks are interesting and important objects to study.!
• We have developed a unified framework that allows a
classification of different types of multilayer networks.!
• Many real networks have multilayer structures.!
• Multilayer networks make it possible to throw away less data.
Additionally, they have interesting structural features and
have interesting effects in dynamical processes.!
• Adjacency tensors: their time has come!
– We need to use tools from multilinear algebra. Tensors generalize
matrices, but there are important differences to consider.!
• Challenge: Need to collect good data, especially w.r.t. realiable
quantitative values for inter-layer edges!
• Challenge: Need more genuinely interlayer diagnostics!
• Not just “bigger and better” version of monoplex objects!
• Challenge: Need additional general results on dynamical processes
(bifurcations, phase transitions). There are some, but we need
more.!
• Challenge: Need to move farther beyond the usual percolation-like
models!
• Not just “bigger and better” versions of monoplex processes!
• Review article of multilayer networks: Journal of Complex Networks,
in press (arXiv:1309.7233)!
• Code for visualization and analysis of multilayer networks:
http://www.plexmath.eu/?page_id=327!
• Thanks: James S. McDonnell Foundation, EPSRC, FET-Proactive project
“PLEXMATH”!

87.
1WVNQQM
• All
is
for
the
best
in
this
best
of
all
possible
worlds.
• (Also:
The
future’s
so
bright,
we
gotta
to
wear
shades.)